Abstract: Quantum technologies are based on the manipulation of individual degrees of freedom of quantum systems with exquisite precision. Achieving this in a real environment requires pushing to the limits the ability to control the dynamics of quantum systems of increasing complexity. Optimal control techniques are known to enable steering the dynamics of few-body systems in order to prepare a desired state or perform a desired unitary transformation. I will present a
recently developed optimal control method that allows doing so for a many-body quantum system undergoing e.g. a quantum phase transition in the non-adiabatic regime. This opens the way to a range of applications, from the suppression of defects in a superfluid-Mott-insulator transition with ultra-cold atoms in an optical lattice to the achievement of various quantum gates at the quantum speed limit. I will present detailed calculations we performed for different
experimental scenario, together with the corresponding results obtained by experimental groups in different fields, from cold atoms to diamond NV centers. Our control method also allows for exploring more general questions like the complexity of reversing quantum many-body dynamics, steering it back to its initial state even without the ability to revert the sign of the whole Hamiltonian. I will conclude by showing some recent results we obtained in this context, as well as further questions opened by our investigations.